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SageMath
E = EllipticCurve("df1")
E.isogeny_class()
Elliptic curves in class 103488.df
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 103488.df1 | 103488x4 | \([0, -1, 0, -125485537, 383682506593]\) | \(14171198121996897746/4077720290568771\) | \(62880443854372908483084288\) | \([2]\) | \(35389440\) | \(3.6567\) | |
| 103488.df2 | 103488x2 | \([0, -1, 0, -115050497, 474966149505]\) | \(21843440425782779332/3100814593569\) | \(23908039794281629679616\) | \([2, 2]\) | \(17694720\) | \(3.3101\) | |
| 103488.df3 | 103488x1 | \([0, -1, 0, -115046577, 475000133553]\) | \(87364831012240243408/1760913\) | \(3394267603550208\) | \([2]\) | \(8847360\) | \(2.9635\) | \(\Gamma_0(N)\)-optimal |
| 103488.df4 | 103488x3 | \([0, -1, 0, -104678177, 564074750625]\) | \(-8226100326647904626/4152140742401883\) | \(-64028043667418530849357824\) | \([2]\) | \(35389440\) | \(3.6567\) |
Rank
sage: E.rank()
The elliptic curves in class 103488.df have rank \(0\).
Complex multiplication
The elliptic curves in class 103488.df do not have complex multiplication.Modular form 103488.2.a.df
sage: E.q_eigenform(10)
Isogeny matrix
sage: E.isogeny_class().matrix()
The \(i,j\) entry is the smallest degree of a cyclic isogeny between the \(i\)-th and \(j\)-th curve in the isogeny class, in the LMFDB numbering.
\(\left(\begin{array}{rrrr} 1 & 2 & 4 & 4 \\ 2 & 1 & 2 & 2 \\ 4 & 2 & 1 & 4 \\ 4 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
